Hubble's Law, when written in this form,
$$
v = H_0D,
$$
means: if $D$ is the current distance of a galaxy, and $H_0$ the Hubble constant, then $v$ is the current recession velocity of the galaxy. So it tells you what the recession velocity of a galaxy is right now, not what it was in the past.
Basically, the Hubble Law is a consequence of the cosmological principle, i.e. that the universe on large scales is isotropic and homogeneous. This means that the expansion of the universe can be described by a single function of time, the so-called scale factor $a(t)$, so that the distance to a faraway galaxy increases over time as
$$
D(t) = a(t)D_c,
$$
where $D_c$ is a constant, called the co-moving distance to the galaxy; $D(t)$ is known as its proper distance. Also, the present-day value of $a(t)$ is set to 1 by convention, i.e. $a(t_0)=1$, so that $D(t_0) = D_c$.
If we take the derivative, then
$$
v(t) = \dot{D}(t) = \dot{a}(t)D_c = \frac{\dot{a}(t)}{a(t)}D(t) = H(t)D(t),
$$
with $v(t)$ called the recession velocity and $H(t)=\dot{a}/a$ the Hubble parameter. This is the general version of Hubble's Law at cosmological time $t$, which at the present day takes the familiar form
$$
v = H_0D,
$$
where $v=v(t_0)$, $H_0=H(t_0)$ and $D=D(t_0)$. But in this form, Hubble's Law isn't very useful: it's a purely theoretical relation, because the recession velocity of a galaxy cannot be directly observed, nor does it say anything about the expansion of the universe in the past. It only tells us how fast a galaxy is moving from us right now, if you know its current distance.
However, there's a related quantity that we can observe, namely the redshift $z$ of a galaxy, which is the change in wavelength of its photons as they travel through the expanding space:
$$
1 + z = \frac{\lambda_\text{ob}}{\lambda_\text{em}},
$$
with $\lambda_\text{em}$, $\lambda_\text{ob}$ the emitted and observed wavelength respectively.
Unlike the recession velocity, the redshift does give us information about the past, because the redshift of a photon accumulates over time, during its journey from the source galaxy to us. By comparing the redshifts of two galaxies, we can deduce information about the expansion rate in the past: suppose we observe two galaxies with distances $D_1 > D_2$ and redshifts $z_1 > z_2$, which emitted their light at times $t_1$, $t_2$ respectively. Then the difference in redshift $z_1-z_2$ will tell you how much the universe expanded in the time interval $[t_1,t_2]$.
In other words, if the expansion of the universe were decelerating, we'd see that the redshift of distant galaxies accumulated a lot in the distant past, when the expansion rate of the universe was high. However, observations showed that the expansion of the universe first decelerated and then started to accelerate again (the transition from deceleration to acceleration occurred when the universe was about 7.7 billion years old). This means that there was a time when the expansion rate was at a minimum, during which the redshift of photons increased less.
The relation between $v$ and $z$ is determined by the cosmological model. In the Standard Model, it can be shown that the observational version of the Hubble Law looks like this:
$$
H_0D = c\int_0^z\frac{\text{d}z'}{\sqrt{\Omega_{R,0}(1+z')^4 + \Omega_{M,0}(1+z')^3 + \Omega_{K,0}(1+z')^2 + \Omega_{\Lambda,0}}},
$$
where $\Omega_{R,0}$, $\Omega_{M,0}$ and $\Omega_{\Lambda,0}$ are the fraction of radiation, matter and dark energy in the present-day universe, and $\Omega_{K,0} = 1 - \Omega_{R,0} - \Omega_{M,0} -\Omega_{\Lambda,0}$ describes the curvature of space.
Early observations and inflation models suggested that the curvature of space is close to zero, which would mean that $\Omega_{M,0}\approx 1$ if there's no dark energy (the contribution of radiation is negligible). On the other hand, dynamical studies of galaxy clusters indicated that $\Omega_{M,0}\approx 0.3$. Furthermore, models without dark energy led to a 'cosmic age' paradox: the calculated age of the universe in these models was less than the age of the oldest observed stars (see Krauss 1995 for a review). These issues were resolved in 1998 when two teams applied Hubble's Law to a sample of supernovae, comparing their distance and redshift, which offered clear evidence for dark energy, with $\Omega_{M,0}\approx 0.3$ and $\Omega_{\Lambda,0}\approx 0.7$, and a Hubble constant $H_0\approx 65\;\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$. These values have been further refined by CMB observations.
The effect of dark energy can be seen in this figure:
Here, I've set $\Omega_{R,0}=0$ and $H_0 = 63.7\;\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$ (the most recently obtained value). The red curve is a model with dark energy. As you can see, for a given distance the corresponding redshift is much lower than in models without dark energy, i.e. without acceleration.
Extra info
It's interesting to examine these models in more detail. Once the values of the cosmological parameters are fixed, the evolution of the universe can be calculated. In particular, the cosmic time can be written as a function of the scale factor:
$$
t(a) = \frac{1}{H_0}\int_0^a\frac{a'\text{d}a'}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a' + \Omega_{K,0}\,a'^2 + \Omega_{\Lambda,0}\,a'^4}},
$$
which can be inverted to yield $a(t)$, and thus also $\dot{a}(t)$ (see also this post and this post). The age of the universe is $t_0=t(1)$, and we find that $t_0 =$ 14.0, 11.8, and 9.7 billion years for $(\Omega_{M,0},\Omega_{\Lambda,0})= (0.3,\,0.7), (0.3,\,0.0), (1.0,\,0.0)$ respectively. In other words, dark energy increases the age of the universe (which also solves the age paradox previously mentioned). This is a crucial point, as I will show below.
Furthermore, there's a simple relation between the redshift of light and the scale factor: if a photon is emitted at a time $t_\text{em}$, then its redshift will accumulate as
$$
z(t) = \frac{a(t)}{a(t_\text{em})} - 1,
$$
so that its present-day observed redshift is $z = 1/a(t_\text{em})-1$ (see wikipedia for a derivation). In other words, the observed redshift of a photon tells us when it was emitted.
Let's apply this to a particular galaxy. Suppose we have a galaxy at a present-day distance $D = 10$ billion lightyears. We then have the following situation:
The first graph shows the proper distance of the galaxy and its light $D(t)=a(t)D$ as a function of lookback time $t_0-t$. In all three cases, $D(t)=0$ corresponds with the 'big bangs' of these models.
The change from dotted lines to solid lines indicate the moment $t_\text{em}$ at which the galaxies emitted the photons that we observe today; the dashed lines are the paths of those photons. In all three models, the photons were emitted about 7 billion years ago. But the corresponding scale factors $a(t_\text{em})$ are very different: $a(t_\text{em})=0.54,\,0.48,\,0.43$ for the red, blue, green models respectively. This is a direct consequence of the different age of the universe in the three cases.
This immediately explains the redshifts shown in the graph below: the present-day redshift of the light is $z=0.86,\,1.1,\,1.3$ in the respective models, i.e. the observed redshift is much lower in the dark energy model.
Although it's not very clear, the red curve of $D(t)$ has an inflection point about 6 billion years ago, corresponding with the moment when $\ddot{a}=0$ and the expansion of the 'red' universe began accelerating. This is much clearer in the top right graph, showing the recession velocity $v(t)=\dot{a}(t)D$. In all three cases, $v(t)$ was much higher in the past, which means that the expansion has been decelerating. But in the dark energy case, $v(t)$ reached a minimum value around 6 billion years ago, and began to increase again. This is the effect of recent acceleration due to dark energy.
However, note that $v(t)$ is much lower in the dark energy universe. Again, this is a consequence of the age of the universe in the models: it took 14 billion years for $a(t)$ to increase from 0 to 1 in the red model, while it took only 9.7 billion years in the green model. As a consequence, $\dot{a}$ is much lower in the former case.
Finally, the last graph shows the Hubble parameter $H(t)=\dot{a}/a$, showing that even in the accelerating universe $H(t)$ decreases.
To summarise, the influence of dark energy determines the redshift, proper distance and the recession velocity over time, but it's not really its effect on the accelerating expansion that's important, but rather its effect on the age of the universe.
As a final note, the proper distance of a galaxy isn't directly observed. It can be derived from its so-called luminosity distance (comparing the apparent brightness and intrinsic luminosities of supernovae). So we should actually compare the evolution of a galaxy with a fixed present-day luminosity distance in different models rather than a fixed proper distance, but this doesn't change the argument.
Forgive me in advance, this may get overcomplicated. I am going to give you the facts as scaled down as I can but still sufficiently detailed. I think providing you with what we have and allowing you to infer from it is the best way to avoid misrepresenting the answer.
Here is the General Relativity equation that describes how gravity interacts with everything else:
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G_NT_{\mu\nu}$$
On the left side is gravity; it is described by terms that have to do with how space curves, expands, and contracts. On the right side is matter, radiation, etc. Pretty much all forms of energy.
It was determined through observation that the universe as we know it is expanding and that the rate of expansion is accelerating. To account for the acceleration of the expansion, the best-fitting theory includes a constant term:
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(R-2\Lambda)=8\pi G_NT_{\mu\nu}$$
Alternatively, one could choose to express that constant term on the right side of the equation. It makes no physical difference. Accordingly, you can interpret this term as a modification to how gravity affects spacetime (if added to the LHS) or as an additional energy term with a negative pressure (if added to the RHS).
Now, as per original GR, we have equations describing the expansion of the universe:
$$\frac{\ddot a}{a}=-\frac{4\pi G_N}{3}\sum_i(\rho_i+3p_i)$$
$a$ represents how much the universe has expanded, $\ddot a$ is the acceleration of the expansion, and $\rho_i$ and $p_i$ are the energy density and the pressure for the $i$th form of energy (the rest are constants). We put in matter, dark matter, radiation, and we even treat the background curvature of the universe as a form of energy here. What we find is that when we treat dark energy as an energy and set its pressure to be equal to negative of its energy density, our equations very closely match the observations. That is sufficient reason to like doing that. However, we also find (via a separate equation) that the energy density of something whose pressure equals the negative of the energy density remains constant for all time. This is unavoidable, it is one of the best fitting models so far.
Furthermore, the constant energy density presents other effects. The energy density of matter or radiation decreases with time. For instance, and this should be intuitive, the energy density of matter decreases like $a^{-3}$. That is, it drops like the cube of the expansion of the universe. And why not? as the universe expands, the volume increases like the expansion cubed; energy density is energy over volume, so it decreases like expansion cubed. Radiation goes like $a^{-4}$, and other energies decrease at varying rates. A constant density means that after a long time, it becomes the dominant term in the above equation. Effectively, after a long time:
$$\frac{\ddot a}{a}=\frac{8\pi G_N}{3}\rho_{DE}$$
This barrage of equations might mean nothing at all to you. That is fine. This is an answer to what dark energy is, why it has a constant energy density, and whether it has gravity.
As for whether or not dark energy can fall into a black hole, that is more complicated. From the point of view of modifying gravity, dark energy is not something that can fall into a black hole. However, from the point of view of being an additional energy term, one might think it must be able to fall into a black hole. Truthfully, I don't know that answer but I know it makes an insignificant difference. Dark energy is too weak to have an effect on the scale of black holes. Even the small gravity of the Sun is enough to negate the effect of dark energy throughout the solar system and probably a bit further. As you can see from the last equation, at late times the acceleration of the expansion is positive. Dark energy never brings the universe to a Big Crunch. For matter, radiation, etc, the acceleration terms at late times all are negative. Dark energy is preventing a Big Crunch.
In response to your last questions. This was never "proven". It was postulated decades ago and has been confirmed by many experiments but none that prove it beyond a doubt. As for what caused the Big Bang. The Big Bang was not an event, it was a moment of time. The Big Bang represents the point in time where the equation for $a$ (remember that's the term that represents how much the universe has expanded) goes to zero. That's it. It has no more need for a cause than has any moment of time after it.
Best Answer
It's not as naive a question as you may think, and the answer is a lot more complicated than you may think.
When we're calculating how the universe expands we assume it's isotropic and homogeneous (this just means on average it's the same everywhere) and it has a scale factor that is normally written as $a(t)$. The scale factor tells us how much the universe has expanded. We set the scale factor to be 1 at the current time, so a scale factor of 2 means everything is twice as far away and a scale factor of 0.5 means everything is half as far away.
If our scale factor $a(t)$ was constant then the universe would be static i.e. distant galaxies would be stationary with respect to us. When we say the universe is expanding we mean that the scale factor $a(t)$ increases with time. To find out how $a(t)$ changes with time we have to solve Einstein's equations, and this is where things get messy because the solutions are complicated. If you're interested have a look at the Wipedia articles on the FLRW metric and the Friedmann equations. Without going into the details, we expect the scale factor to look something like:
Without dark energy the rate of increase of the scale factor gradually decreases with time because the mutual gravitational attraction of all the matter in the universe slows the expansion. With dark energy the expansion rate is always slightly higher, and at large times the expansion rate starts increasing again.
The original detection of dark energy was based on measuring the recession velocity of supernovae, and discovering that they matched the predictions from the red line not the black one.
A few additional notes: we believe the universe is flat, and this means that (in the absence of dark energy) the expansion rate shown by the black line would continue to slow but would never actually become zero. More precisely it would tend asymptotically to zero as time tends to infinity. An open universe means the expansion rate would tend to a value greater than zero, and a closed universe means the expansion rate would reach zero in finite time then become negative. A closed universe would start contracting again.
Also note that at time zero the scale factor is zero i.e. the distance between everything in the universe would be zero. This point is what we call the Big Bang. The name is misleading because it wasn't an explosion (as so often shown on popular science TV programmes). It's actually a singularity because if the spacing between everything is zero the density must be infinite. A singular point is where our equations break down because we can't do arithmetic with the number $\infty$.